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Groups > sci.physics.relativity > #376988 > unrolled thread
| Started by | JanPB <filmart@gmail.com> |
|---|---|
| First post | 2016-02-23 17:05 -0800 |
| Last post | 2016-02-26 15:02 -0800 |
| Articles | 20 on this page of 134 — 18 participants |
Back to article view | Back to sci.physics.relativity
Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-23 17:05 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-23 22:28 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-23 22:47 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-23 23:01 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-23 23:04 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-23 23:27 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-23 23:28 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-23 23:40 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-24 00:21 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Maciej Woźniak <mlwozniak@wp.pl> - 2016-02-24 10:39 +0100
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-24 12:59 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-24 13:12 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-24 08:46 -0600
Re: Schwarzschild's paper and the uniqueness of the solution "Paul B. Andersen" <relativity@paulba.no> - 2016-02-24 19:54 +0100
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-24 01:20 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-24 01:39 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Maciej Woźniak <mlwozniak@wp.pl> - 2016-02-24 11:12 +0100
Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-24 08:37 -0600
Re: Schwarzschild's paper and the uniqueness of the solution "Dono," <sa_ge@comcast.net> - 2016-02-24 07:16 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-24 22:31 -0800
Re: Schwarzschild's paper and the uniqueness of the solution "Dono," <sa_ge@comcast.net> - 2016-02-25 06:30 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 09:11 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-25 16:13 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 09:38 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-25 17:44 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 09:50 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-25 11:51 -0600
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 09:57 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-25 12:05 -0600
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 10:19 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-25 18:13 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 10:21 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-25 18:05 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-25 09:48 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Jackpol11@hotmail.com - 2016-02-25 14:06 -0500
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 12:35 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-25 21:44 +0000
Re: Schwarzschild's paper and the uniqueness of the solution Jackpol11@hotmail.com - 2016-02-25 17:37 -0500
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 15:11 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 12:44 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-26 06:57 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 14:59 +0000
Re: Schwarzschild's paper and the uniqueness of the solution Gary Harnagel <hitlong@yahoo.com> - 2016-02-26 07:36 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 18:45 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-26 07:37 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 19:36 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-26 07:50 -0800
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 18:47 +0000
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 11:44 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-25 15:47 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 16:00 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-25 16:09 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Odd Bodkin <bodkinodd@gmail.com> - 2016-02-25 18:29 -0600
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-25 18:43 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-26 23:04 +0100
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 14:59 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-02-26 16:19 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-27 10:27 +0100
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-27 17:55 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 12:13 +0100
Re: Schwarzschild's paper and the uniqueness of the solution Emigdio Sophronius <emigg@augustvisitors.org> - 2016-02-28 11:37 +0000
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 12:56 +0100
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 08:03 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:13 +0100
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:26 +0100
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 09:36 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:50 +0100
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 09:54 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 20:40 +0100
Re: Schwarzschild's paper and the uniqueness of the solution Emigdio Sophronius <emigg@augustvisitors.org> - 2016-02-28 19:44 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 12:20 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Emigdio Sophronius <emigg@augustvisitors.org> - 2016-02-28 20:58 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 13:02 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 12:31 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-29 22:40 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 10:54 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 07:30 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 17:46 +0100
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 08:51 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Emigdio Sophronius <emigg@augustvisitors.org> - 2016-02-28 17:01 +0000
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 07:34 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 15:15 -0600
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-03-03 19:54 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 08:48 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Fuller <fuller.david@hotmail.com> - 2016-02-28 09:13 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 10:09 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Fuller <fuller.david@hotmail.com> - 2016-03-06 10:56 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:19 +0100
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 09:24 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 18:34 +0100
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 09:44 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-02-28 19:12 +0100
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 10:19 -0800
Re: Schwarzschild's paper and the uniqueness of the solution David Waite <waitedavid1618@yahoo.com> - 2016-02-28 11:17 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-01 17:06 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-03-01 17:54 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-01 19:41 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-03-04 01:17 +0100
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-03 17:12 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-03-03 17:53 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 22:19 -0600
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-03 21:36 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-10 11:14 -0600
Re: Schwarzschild's paper and the uniqueness of the solution Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-10 18:32 +0100
Re: Schwarzschild's paper and the uniqueness of the solution Luigi Durante <luigdu@jaccappo.org> - 2016-03-10 20:22 +0000
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-10 13:02 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Luigi Durante <luigdu@jaccappo.org> - 2016-03-10 21:08 +0000
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-10 13:19 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Luigi Durante <luigdu@jaccappo.org> - 2016-03-10 21:27 +0000
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-10 14:04 -0800
Re: Schwarzschild's paper and the uniqueness of the solution "Paul B. Andersen" <relativity@paulba.no> - 2016-03-11 13:33 +0100
Re: Schwarzschild's paper and the uniqueness of the solution Maciej Woźniak <mlwozniak@wp.pl> - 2016-03-10 22:49 +0100
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-10 14:20 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-03-05 05:24 +0100
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-04 21:36 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-03-04 22:50 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Thomas 'PointedEars' Lahn <PointedEars@web.de> - 2016-03-05 11:24 +0100
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-05 20:07 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 14:49 -0600
Re: Schwarzschild's paper and the uniqueness of the solution Helénē Swanhild <helenes@gardenclosure.org> - 2016-03-03 21:40 +0000
Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 18:58 -0600
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-03-03 14:04 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Koobee Wublee <koobee.wublee@gmail.com> - 2016-03-03 17:46 -0800
Re: Schwarzschild's paper and the uniqueness of the solution Tom Roberts <tjroberts137@sbcglobal.net> - 2016-03-03 22:21 -0600
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-25 16:42 -0800
Re: Schwarzschild's paper and the uniqueness of the solution "Paul B. Andersen" <relativity@paulba.no> - 2016-02-26 20:00 +0100
Re: Schwarzschild's paper and the uniqueness of the solution PC <pc468@hotmail.com> - 2016-02-26 19:03 +0000
Re: Schwarzschild's paper and the uniqueness of the solution John Gogo <jfgogo22@yahoo.com> - 2016-02-25 19:36 -0800
Re: Schwarzschild's paper and the uniqueness of the solution alsor@interia.pl - 2016-02-26 09:36 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 11:33 -0800
Re: Schwarzschild's paper and the uniqueness of the solution alsor@interia.pl - 2016-02-26 12:05 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 12:47 -0800
Re: Schwarzschild's paper and the uniqueness of the solution alsor@interia.pl - 2016-02-26 14:16 -0800
Re: Schwarzschild's paper and the uniqueness of the solution JanPB <filmart@gmail.com> - 2016-02-26 15:02 -0800
Page 6 of 7 — ← Prev page 1 2 3 4 5 [6] 7 Next page →
| From | Tom Roberts <tjroberts137@sbcglobal.net> |
|---|---|
| Date | 2016-03-03 22:19 -0600 |
| Message-ID | <t-udnYxLEspHkUTLnZ2dnUU7_83NnZ2d@giganews.com> |
| In reply to | #377846 |
On 3/3/16 3/3/16 7:12 PM, JanPB wrote: > On Thursday, March 3, 2016 at 4:17:02 PM UTC-8, Thomas 'PointedEars' Lahn wrote: >> [about ds^2] > > You said it was a scalar. So it must be a well-defined number at each point > of the sphere. He said it is a scalar, not a scalar FIELD on the sphere. In the physicists' usage of the line element, ds^2 for any given pair of infinitesimally-separated points is indeed a scalar (i.e. has the same value when computed in any coordinate system). It is definitely NOT a scalar field. You two are trying very hard to find a disagreement and argument, when you actually agree on all the major points of geometry and physics; only your notations and nomenclature are different. Tom Roberts
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| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2016-03-03 21:36 -0800 |
| Message-ID | <e268942d-03e8-4d43-aeb2-1bf8aaf9ad3a@googlegroups.com> |
| In reply to | #377855 |
On Thursday, March 3, 2016 at 8:19:41 PM UTC-8, tjrob137 wrote: > On 3/3/16 3/3/16 7:12 PM, JanPB wrote: > > On Thursday, March 3, 2016 at 4:17:02 PM UTC-8, Thomas 'PointedEars' Lahn wrote: > >> [about ds^2] > > > > You said it was a scalar. So it must be a well-defined number at each point > > of the sphere. > > He said it is a scalar, not a scalar FIELD on the sphere. > > In the physicists' usage of the line element, ds^2 for any given pair of > infinitesimally-separated points is indeed a scalar (i.e. has the same value > when computed in any coordinate system). It is definitely NOT a scalar field. Problem is, there is no such thing as "pair of infinitesimally-separated points". It's either a metaphor or a tangent vector. And, of course, ds^2 evaluated at a tangent vector is a number. > You two are trying very hard to find a disagreement and argument, when you > actually agree on all the major points of geometry and physics; only your > notations and nomenclature are different. Only trying to point out that the "infinitesimal" concept is ill-defined (always has been) and today (well, for many decades now) there is no need to insist on its literal meaning. -- Jan
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| From | Tom Roberts <tjroberts137@sbcglobal.net> |
|---|---|
| Date | 2016-03-10 11:14 -0600 |
| Message-ID | <0aKdnTBtXYv6NnzLnZ2dnUU7_8zNnZ2d@giganews.com> |
| In reply to | #377861 |
On 3/3/16 3/3/16 11:36 PM, JanPB wrote: > On Thursday, March 3, 2016 at 8:19:41 PM UTC-8, tjrob137 wrote: >> On 3/3/16 3/3/16 7:12 PM, JanPB wrote: >>> On Thursday, March 3, 2016 at 4:17:02 PM UTC-8, Thomas 'PointedEars' Lahn wrote: >>>> [about ds^2] >>> >>> You said it was a scalar. So it must be a well-defined number at each point >>> of the sphere. >> >> He said it is a scalar, not a scalar FIELD on the sphere. >> >> In the physicists' usage of the line element, ds^2 for any given pair of >> infinitesimally-separated points is indeed a scalar (i.e. has the same value >> when computed in any coordinate system). It is definitely NOT a scalar field. > > Problem is, there is no such thing as "pair of infinitesimally-separated points". It's > either a metaphor or a tangent vector. And, of course, ds^2 evaluated at a tangent > vector is a number. Yes, mathematicians are concerned with such issues. Physicists are not, and the issues you raise are not important in physics, because our manifolds are all well behaved and we don't insist on strict mathematical rigor. The usual line element _CAN_ be used to integrate distance along a line (path), and it _CAN_ be used to read off the metric components in the coordinates used. (IMHO this last is its primary use: as a succinct statement of the metric components.) Physicists' use of the line element became established well before the issues you raise were generally recognized. There's no need for physicists' usage of this to change, though it is certainly possible to use it as you say. Tom Roberts
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| From | Maciej Woźniak <mlwozniak@wp.pl> |
|---|---|
| Date | 2016-03-10 18:32 +0100 |
| Message-ID | <nbsb4j$48g$1@node1.news.atman.pl> |
| In reply to | #378486 |
Użytkownik "Tom Roberts" napisał w wiadomości grup dyskusyjnych:0aKdnTBtXYv6NnzLnZ2dnUU7_8zNnZ2d@giganews.com... |Yes, mathematicians are concerned with such issues. |Physicists are not, and the issues you raise are not important in physics, |because Because common sense is a set of prejudices!!!!! Yahoooooo!!!!!
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| From | Luigi Durante <luigdu@jaccappo.org> |
|---|---|
| Date | 2016-03-10 20:22 +0000 |
| Message-ID | <nbsl1h$1reb$1@gioia.aioe.org> |
| In reply to | #378486 |
Tom Roberts wrote: > On 3/3/16 3/3/16 11:36 PM, JanPB wrote: >> Problem is, there is no such thing as "pair of >> infinitesimally-separated points". It's either a metaphor or a tangent >> vector. And, of course, ds^2 evaluated at a tangent vector is a number. > > Yes, mathematicians are concerned with such issues. > Physicists are not, and the issues you raise are not important in > physics, because our manifolds are all well behaved and we don't insist > on strict mathematical rigor. The usual line element _CAN_ be used to > integrate distance along a line (path), and it _CAN_ be used to read off > the metric components in the coordinates used. (IMHO this last is its > primary use: as a succinct statement of the metric components.) > Physicists' use of the line element became established well before the > issues you raise were generally recognized. There's no need for > physicists' usage of this to change, though it is certainly possible to > use it as you say. Tom Roberts You are too kind, mister Tom, not saying he talks nonsense. As he thinks that two infinitesimally separated points on a sphere are connected by a tangent, lol. A mathematical impossibility, no? Ok.
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| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2016-03-10 13:02 -0800 |
| Message-ID | <632cbe96-46cf-4646-ac83-7353c1262995@googlegroups.com> |
| In reply to | #378504 |
On Thursday, March 10, 2016 at 12:22:13 PM UTC-8, Luigi Durante wrote: > Tom Roberts wrote: > > > On 3/3/16 3/3/16 11:36 PM, JanPB wrote: > >> Problem is, there is no such thing as "pair of > >> infinitesimally-separated points". It's either a metaphor or a tangent > >> vector. And, of course, ds^2 evaluated at a tangent vector is a number. > > > > Yes, mathematicians are concerned with such issues. > > Physicists are not, and the issues you raise are not important in > > physics, because our manifolds are all well behaved and we don't insist > > on strict mathematical rigor. The usual line element _CAN_ be used to > > integrate distance along a line (path), and it _CAN_ be used to read off > > the metric components in the coordinates used. (IMHO this last is its > > primary use: as a succinct statement of the metric components.) > > Physicists' use of the line element became established well before the > > issues you raise were generally recognized. There's no need for > > physicists' usage of this to change, though it is certainly possible to > > use it as you say. Tom Roberts > > You are too kind, mister Tom, not saying he talks nonsense. As he thinks > that two infinitesimally separated points on a sphere are connected by a > tangent, lol. A mathematical impossibility, no? Ok. Newsflash: that's not what I said. What I said was: 1. "two infinitesimally separated points" is a metaphor, it's not valid mathematics. That metaphor is _very_ useful provided you have enough experience to know how to work with it correctly, 2. the legal, correct, mathematical object that represents that metaphor is the tangent vector. That vector obviously does not "connect" any "two points" on the sphere. -- Jan
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| From | Luigi Durante <luigdu@jaccappo.org> |
|---|---|
| Date | 2016-03-10 21:08 +0000 |
| Message-ID | <nbsnor$1vpc$1@gioia.aioe.org> |
| In reply to | #378512 |
JanPB wrote: >> You are too kind, mister Tom, not saying he talks nonsense. As he >> thinks that two infinitesimally separated points on a sphere are >> connected by a tangent, lol. A mathematical impossibility, no? Ok. > > Newsflash: that's not what I said. What I said was: > > 1. "two infinitesimally separated points" is a metaphor, it's not valid > mathematics. That metaphor is _very_ useful provided you have enough > experience to know how to work with it correctly, > > 2. the legal, correct, mathematical object that represents that metaphor > is the tangent vector. That vector obviously does not "connect" any "two > points" on the sphere. Apparently you contradict yourself again. 1. "two infinitesimally separated points" is a metaphor, 2. that metaphor is the tangent vector 3. "two infinitesimally separated points" = "the tangent vector" Well, how not? Or is us incapable to read and simplify, no? Ok.
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| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2016-03-10 13:19 -0800 |
| Message-ID | <e59a8c76-fe79-4f31-8864-63d68ad5646b@googlegroups.com> |
| In reply to | #378513 |
On Thursday, March 10, 2016 at 1:08:47 PM UTC-8, Luigi Durante wrote: > JanPB wrote: > > >> You are too kind, mister Tom, not saying he talks nonsense. As he > >> thinks that two infinitesimally separated points on a sphere are > >> connected by a tangent, lol. A mathematical impossibility, no? Ok. > > > > Newsflash: that's not what I said. What I said was: > > > > 1. "two infinitesimally separated points" is a metaphor, it's not valid > > mathematics. That metaphor is _very_ useful provided you have enough > > experience to know how to work with it correctly, > > > > 2. the legal, correct, mathematical object that represents that metaphor > > is the tangent vector. That vector obviously does not "connect" any "two > > points" on the sphere. > > Apparently you contradict yourself again. > > 1. "two infinitesimally separated points" is a metaphor, > 2. that metaphor is the tangent vector > > 3. "two infinitesimally separated points" = "the tangent vector" > > Well, how not? Or is us incapable to read and simplify, no? Ok. Looks like you are more interesting in arguing than actually reading what I wrote. OK, one more time (only): Your (1) above is a correct interpretation of what I wrote. (2) and (3) are not. I said tangent vector represents the metaphor (i.e., its an actual mathematical object which behaves exactly as one would _expect_ the metaphor to behave (if it was a real object)). And of course there is no such equality as (3) because the LHS doesn't exist (it's a metaphor). -- Jan
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| From | Luigi Durante <luigdu@jaccappo.org> |
|---|---|
| Date | 2016-03-10 21:27 +0000 |
| Message-ID | <nbsosb$tb$3@gioia.aioe.org> |
| In reply to | #378516 |
JanPB wrote: > On Thursday, March 10, 2016 at 1:08:47 PM UTC-8, Luigi Durante wrote: >> JanPB wrote: >> >> >> You are too kind, mister Tom, not saying he talks nonsense. As he >> >> thinks that two infinitesimally separated points on a sphere are >> >> connected by a tangent, lol. A mathematical impossibility, no? Ok. >> > >> > Newsflash: that's not what I said. What I said was: >> > >> > 1. "two infinitesimally separated points" is a metaphor, it's not >> > valid mathematics. That metaphor is _very_ useful provided you have >> > enough experience to know how to work with it correctly, >> > >> > 2. the legal, correct, mathematical object that represents that >> > metaphor is the tangent vector. That vector obviously does not >> > "connect" any "two points" on the sphere. >> >> Apparently you contradict yourself again. >> >> 1. "two infinitesimally separated points" is a metaphor, >> 2. that metaphor is the tangent vector >> >> 3. "two infinitesimally separated points" = "the tangent vector" >> >> Well, how not? Or is us incapable to read and simplify, no? Ok. > > Looks like you are more interesting in arguing than actually reading > what I wrote. OK, one more time (only): Not at all, you feels cornered. Admit it and let's talk about something else > > Your (1) above is a correct interpretation of what I wrote. > (2) and (3) are not. 2. ".. that metaphor is the tangent vector." 3. is perfect, mathematically. > I said tangent vector represents the metaphor (i.e., its an actual > mathematical object which behaves exactly as one would _expect_ the > metaphor to behave (if it was a real object)). > And of course there is no such equality as (3) because the LHS doesn't > exist (it's a metaphor). It is, not? Ok. WORLD EXCLUSIVE-DAVID CAMERON TO COME OUT AS TRANSGENDER https://www.youtube.com/watch?v=dnRZjb4GkDE
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| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2016-03-10 14:04 -0800 |
| Message-ID | <66c9713f-ef59-4ff6-a03b-6c469299b902@googlegroups.com> |
| In reply to | #378518 |
On Thursday, March 10, 2016 at 1:27:43 PM UTC-8, Luigi Durante wrote: > JanPB wrote: > > > On Thursday, March 10, 2016 at 1:08:47 PM UTC-8, Luigi Durante wrote: > >> JanPB wrote: > >> > >> >> You are too kind, mister Tom, not saying he talks nonsense. As he > >> >> thinks that two infinitesimally separated points on a sphere are > >> >> connected by a tangent, lol. A mathematical impossibility, no? Ok. > >> > > >> > Newsflash: that's not what I said. What I said was: > >> > > >> > 1. "two infinitesimally separated points" is a metaphor, it's not > >> > valid mathematics. That metaphor is _very_ useful provided you have > >> > enough experience to know how to work with it correctly, > >> > > >> > 2. the legal, correct, mathematical object that represents that > >> > metaphor is the tangent vector. That vector obviously does not > >> > "connect" any "two points" on the sphere. > >> > >> Apparently you contradict yourself again. > >> > >> 1. "two infinitesimally separated points" is a metaphor, > >> 2. that metaphor is the tangent vector > >> > >> 3. "two infinitesimally separated points" = "the tangent vector" > >> > >> Well, how not? Or is us incapable to read and simplify, no? Ok. > > > > Looks like you are more interesting in arguing than actually reading > > what I wrote. OK, one more time (only): > > Not at all, you feels cornered. I'm glad I defeated you. -- Jan
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| From | "Paul B. Andersen" <relativity@paulba.no> |
|---|---|
| Date | 2016-03-11 13:33 +0100 |
| Message-ID | <nbudpi$ht9$1@dont-email.me> |
| In reply to | #378521 |
On 10.03.2016 23:04, JanPB wrote: > On Thursday, March 10, 2016 at 1:27:43 PM UTC-8, Luigi Durante wrote: >> >> Not at all, you feels cornered. > > I'm glad I defeated you. > > -- > Jan > Why do you take the bait of this trolling nym-shifter? Wasn't the line too obvious to miss? -- Paul https://paulba.no/
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| From | Maciej Woźniak <mlwozniak@wp.pl> |
|---|---|
| Date | 2016-03-10 22:49 +0100 |
| Message-ID | <nbsq69$kmg$1@node1.news.atman.pl> |
| In reply to | #378512 |
Użytkownik "JanPB" napisał w wiadomości grup dyskusyjnych:632cbe96-46cf-4646-ac83-7353c1262995@googlegroups.com... > > Yes, mathematicians are concerned with such issues. > > Physicists are not, and the issues you raise are not important in > > physics, because our manifolds are all well behaved and we don't insist > > on strict mathematical rigor. The usual line element _CAN_ be used to > > integrate distance along a line (path), and it _CAN_ be used to read off > > the metric components in the coordinates used. (IMHO this last is its > > primary use: as a succinct statement of the metric components.) > > Physicists' use of the line element became established well before the > > issues you raise were generally recognized. There's no need for > > physicists' usage of this to change, though it is certainly possible to > > use it as you say. Tom Roberts > > You are too kind, mister Tom, not saying he talks nonsense. As he thinks > that two infinitesimally separated points on a sphere are connected by a > tangent, lol. A mathematical impossibility, no? Ok. |Newsflash: that's not what I said. What I said was: |1. "two infinitesimally separated points" is a metaphor, it's not valid |mathematics. That metaphor is _very_ useful provided you have enough |experience to know how to work with it correctly, A lie, as usual from relativistic trash. What you really said, is: > Problem is, there is no such thing as "pair of infinitesimally-separated > points".
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| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2016-03-10 14:20 -0800 |
| Message-ID | <1a880906-be58-4831-982e-1e4955fbfbae@googlegroups.com> |
| In reply to | #378520 |
On Thursday, March 10, 2016 at 1:50:03 PM UTC-8, Maciej Woźniak wrote: > Użytkownik "JanPB" napisał w wiadomości grup > dyskusyjnych:632cbe96-46cf-4646-ac83-7353c1262995@googlegroups.com... > > > > > Yes, mathematicians are concerned with such issues. > > > Physicists are not, and the issues you raise are not important in > > > physics, because our manifolds are all well behaved and we don't insist > > > on strict mathematical rigor. The usual line element _CAN_ be used to > > > integrate distance along a line (path), and it _CAN_ be used to read off > > > the metric components in the coordinates used. (IMHO this last is its > > > primary use: as a succinct statement of the metric components.) > > > Physicists' use of the line element became established well before the > > > issues you raise were generally recognized. There's no need for > > > physicists' usage of this to change, though it is certainly possible to > > > use it as you say. Tom Roberts > > > > You are too kind, mister Tom, not saying he talks nonsense. As he thinks > > that two infinitesimally separated points on a sphere are connected by a > > tangent, lol. A mathematical impossibility, no? Ok. > > |Newsflash: that's not what I said. What I said was: > |1. "two infinitesimally separated points" is a metaphor, it's not valid > |mathematics. That metaphor is _very_ useful provided you have enough > |experience to know how to work with it correctly, > > A lie, as usual from relativistic trash. > What you really said, is: > > Problem is, there is no such thing as "pair of infinitesimally-separated > > points". You are a moron. -- Jan
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| From | Thomas 'PointedEars' Lahn <PointedEars@web.de> |
|---|---|
| Date | 2016-03-05 05:24 +0100 |
| Message-ID | <56DA5F8C.7010402@PointedEars.de> |
| In reply to | #377846 |
JanPB wrote:
> On Thursday, March 3, 2016 at 4:17:02 PM UTC-8, Thomas 'PointedEars'
> Lahn wrote:
>> JanPB wrote:
>>> On Sunday, February 28, 2016 at 3:13:08 AM UTC-8, Thomas
>>> 'PointedEars' Lahn wrote:
>>
>> Attribution _line_, not attribution novel.
Which part of that did you not understand?
>>>> JanPB wrote:
>>>>> [...] Thomas 'PointedEars' Lahn wrote:
>>>>> […] The fact is that ds^2 is a tensor, not a number.
>>>> ^^^^^^^^^^^^
>>>> How did you get that idea?
>>> It's the definition.
>> Proof by assertion.
> Sorry, this just is the definition of ds^2. I cannot change it.
We have already determined that there is not one single correct definition,
so it is not *the* definition, but *a* definition (that you are using).
>>> You don't believe me?
>> No, as always, I have to see evidence supporting your opinion.
> Pick any book on differential geometry (not an undergraduate hand-waving
> text though), e.g.:
>
> Barrett O'Neill: "Semi-Riemannian Geometry",
> […]
> etc.
Insufficient. You need to *cite* your sources to substantiate your claims.
Citing means referring to the *exact* statement that does this. (If you are
who you claim you are, you know this already.)
> You said it was a scalar. So it must be a well-defined number at each
> point of the sphere. What number is it? What's the formula? Be precise.
Your logic is still flawed.
>>>> But "ds" alone does _not_ define the geometry;
>>> Since we differ what the symbol "ds" means, we won't agree here.
>> Non sequitur.
> So now we DO agree what the symbol "ds" means?
No, we do not. For one, I am willing to accept the possibility that it does
not have only one meaning, while you do not.
> I'll take your non-sequitur and raise it to falsehood then :-) Make up
> your mind!
I have not changed my mind. Look up the meaning of “non sequitur”. (Again,
if you are who you claim you are, you should not have to.)
>> But *that* [ds] "*defines* the geometry" is only your idea. So whether
>> your preference in terminology is justified is an open issue at this
>> point.
> It's not open to debate.
Apparently it is.
> […] in the context of _differential geometry_. There exist different
> concepts of "geometry", some of them not related to calculus on
> manifolds at all. They are not relevant to GR.
We are not discussing GR, and as for “differential geometry”, you have still
not substantiated that.
>>> […] The phrase "ds^2 is just a scalar that represents the
>>> geometry" is wrong: it's not a scalar, it's a rank-2 tensor.
>> As for that, I refer you to Tom Robert's excellent explanation in the
>> upper part¹ of <news:tfKdndZl7qz9PkXLnZ2dnUU7_8zNnZ2d@giganews.com> ,
>> and rest my case. (We are both correct in our respective
>> interpretations.)
> I disagree because you still haven't shown me any viable alternative
Why should I repeat what Tom said? Can’t you read?
> except metaphors like "infinitesimal distance" and the like.
If you cared to read *properly*, I have written (and quoted) “infinitesimal
(infinitely small) change”.
>>> It's a scalar only in the sense that when it's operating of a pair of
>> ^^^^^^^^^^^^^^^^^^^^^^
>>> vectors, it [...]
>> ^^^^^^^
>> What does that mean?
> Good grief. OK, I'll stop here.
Posting gibberish? That would be advisable indeed. Post in proper language
then. “it’s operating of a pair of vectors” is _not_.
(You did use a fixed-width font for display as it is advisable in Usenet —
to understand those markings, for example –, yes?)
PointedEars
.
.
.
.
.
.
.
--
“Science is empirical: knowing the answer means nothing;
testing your knowledge means everything.”
—Dr. Lawrence M. Krauss, theoretical physicist,
in “A Universe from Nothing” (2009)
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| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2016-03-04 21:36 -0800 |
| Message-ID | <9a29e27b-001a-4c49-9c4a-7a112b24ee2f@googlegroups.com> |
| In reply to | #377923 |
On Friday, March 4, 2016 at 8:24:45 PM UTC-8, Thomas 'PointedEars' Lahn wrote: > JanPB wrote: > > > On Thursday, March 3, 2016 at 4:17:02 PM UTC-8, Thomas 'PointedEars' > > Lahn wrote: > >> JanPB wrote: > >>> On Sunday, February 28, 2016 at 3:13:08 AM UTC-8, Thomas > >>> 'PointedEars' Lahn wrote: > >> > >> Attribution _line_, not attribution novel. > > Which part of that did you not understand? I didn't write this. > >>>> JanPB wrote: > >>>>> [...] Thomas 'PointedEars' Lahn wrote: > >>>>> [...] The fact is that ds^2 is a tensor, not a number. > >>>> ^^^^^^^^^^^^ > > >>>> How did you get that idea? > >>> It's the definition. > >> Proof by assertion. > > Sorry, this just is the definition of ds^2. I cannot change it. > > We have already determined that there is not one single correct definition, No, we haven't determined that. > so it is not *the* definition, but *a* definition (that you are using). It is *the* definition. There is no other one. Everything else you might hear are metaphors. Again, I'm not saying metaphors are bad or useless, only that they are not definitions. > >>> You don't believe me? > >> No, as always, I have to see evidence supporting your opinion. > > Pick any book on differential geometry (not an undergraduate hand-waving > > text though), e.g.: > > > > Barrett O'Neill: "Semi-Riemannian Geometry", > > [...] > > etc. > > Insufficient. Sufficient. > You need to *cite* your sources to substantiate your claims. No. > Citing means referring to the *exact* statement that does this. (If you are > who you claim you are, you know this already.) No. Just look up the references. > > You said it was a scalar. So it must be a well-defined number at each > > point of the sphere. What number is it? What's the formula? Be precise. > > Your logic is still flawed. No. Answer the question. > >>>> But "ds" alone does _not_ define the geometry; > >>> Since we differ what the symbol "ds" means, we won't agree here. > >> Non sequitur. > > So now we DO agree what the symbol "ds" means? > > No, we do not. For one, I am willing to accept the possibility that it does > not have only one meaning, while you do not. ds^2 has only one meaning as far as the actual definition of this object is concerned. Again, I'm not denying that useful paraphrases exist. [...] > >> But *that* [ds] "*defines* the geometry" is only your idea. So whether > >> your preference in terminology is justified is an open issue at this > >> point. > > It's not open to debate. > > Apparently it is. No. > > [...] in the context of _differential geometry_. There exist different > > concepts of "geometry", some of them not related to calculus on > > manifolds at all. They are not relevant to GR. > > We are not discussing GR, and as for "differential geometry", you have still > not substantiated that. We are not discussing GR? That's news to me. What are we discussing then? Also, there is nothing to "substantiate", it's what ds^2 is. [...] > > except metaphors like "infinitesimal distance" and the like. > > If you cared to read *properly*, I have written (and quoted) "infinitesimal > (infinitely small) change". It's a metaphor. Nothing wrong with that, just keep in mind it doesn't define anything. One can get quite far using only metaphors (see Newton!) but at some point one reaches a point of crisis. > >>> It's a scalar only in the sense that when it's operating of a pair of > >> ^^^^^^^^^^^^^^^^^^^^^^ > >>> vectors, it [...] > >> ^^^^^^^ > >> What does that mean? > > Good grief. OK, I'll stop here. > > Posting gibberish? That would be advisable indeed. Post in proper language > then. "it's operating of a pair of vectors" is _not_. No. If you don't know what I'm saying here then I'll stop because I cannot type in an entire math class here. -- Jan
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| From | Koobee Wublee <koobee.wublee@gmail.com> |
|---|---|
| Date | 2016-03-04 22:50 -0800 |
| Message-ID | <627d15ea-0b23-4c0e-b95c-ede8f6da3087@googlegroups.com> |
| In reply to | #377927 |
On Friday, March 4, 2016 at 9:36:48 PM UTC-8, JanPB wrote: > If you don't know what I'm saying here then I'll stop because I > cannot type in an entire math class here. You don't have to type in the entire math class. There are plenty of elementary books to teach you. You might find the following list of books educational in helping you to understand what equality means in math. <shrug> https://www.google.com/?gws_rd=ssl#q=2nd+grade+math+textbook&tbm=shop&spd=14442141523518876256
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| From | Thomas 'PointedEars' Lahn <PointedEars@web.de> |
|---|---|
| Date | 2016-03-05 11:24 +0100 |
| Message-ID | <56DAB3E8.3010808@PointedEars.de> |
| In reply to | #377927 |
JanPB wrote on 05.03.2016 06:36: > On Friday, March 4, 2016 at 8:24:45 PM UTC-8, Thomas 'PointedEars' Lahn wrote: >> JanPB wrote: >>> On Thursday, March 3, 2016 at 4:17:02 PM UTC-8, Thomas 'PointedEars' >>> Lahn wrote: >>>> JanPB wrote: >>>>> On Sunday, February 28, 2016 at 3:13:08 AM UTC-8, Thomas >>>>> 'PointedEars' Lahn wrote: >>>> Attribution _line_, not attribution novel. >> Which part of that did you not understand? > > I didn't write this. Of course not, I did, implicitly asking you to trim your attributions so that they do not exceed one line (and are making discussions in which you participate harder to read). Can you not even discern quotation levels now? >>>>>> JanPB wrote: >>>>>>> [...] Thomas 'PointedEars' Lahn wrote: >>>>>>> [...] The fact is that ds^2 is a tensor, not a number. >>>>>> ^^^^^^^^^^^^ >> >>>>>> How did you get that idea? >>>>> It's the definition. >>>> Proof by assertion. >>> Sorry, this just is the definition of ds^2. I cannot change it. >> >> We have already determined that there is not one single correct definition, > > No, we haven't determined that. Well, we, Tom Roberts and I at least, have. You just do not accept the fact. A very basic fact at that: for every term there can be more than one definition. Until you do it is a waste of time to continue discussing this with you. -- PointedEars
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| From | JanPB <filmart@gmail.com> |
|---|---|
| Date | 2016-03-05 20:07 -0800 |
| Message-ID | <ea1c5c3d-1a75-44dc-951c-a1b31fe4b667@googlegroups.com> |
| In reply to | #377934 |
On Saturday, March 5, 2016 at 2:24:43 AM UTC-8, Thomas 'PointedEars' Lahn wrote: > JanPB wrote on 05.03.2016 06:36: > > On Friday, March 4, 2016 at 8:24:45 PM UTC-8, Thomas 'PointedEars' Lahn wrote: > >> JanPB wrote: > >>> On Thursday, March 3, 2016 at 4:17:02 PM UTC-8, Thomas 'PointedEars' > >>> Lahn wrote: > >>>> JanPB wrote: > >>>>> On Sunday, February 28, 2016 at 3:13:08 AM UTC-8, Thomas > >>>>> 'PointedEars' Lahn wrote: > >>>> Attribution _line_, not attribution novel. > >> Which part of that did you not understand? > > > > I didn't write this. > > Of course not, I did, implicitly asking you to trim your attributions so > that they do not exceed one line (and are making discussions in which you > participate harder to read). Can you not even discern quotation levels now? I don't know what you are talking about. > >>>>>> JanPB wrote: > >>>>>>> [...] Thomas 'PointedEars' Lahn wrote: > >>>>>>> [...] The fact is that ds^2 is a tensor, not a number. > >>>>>> ^^^^^^^^^^^^ > >> > >>>>>> How did you get that idea? > >>>>> It's the definition. > >>>> Proof by assertion. > >>> Sorry, this just is the definition of ds^2. I cannot change it. > >> > >> We have already determined that there is not one single correct definition, > > > > No, we haven't determined that. > > Well, we, Tom Roberts and I at least, have. You can agree all you want, it still won't make infinitesimals into legal mathematical objects (unless you are talking about non-standard analysis - let me know if you do so I can stop pointing this out). > You just do not accept the fact. What fact? Infinitesimals do not exist, they are only metaphors (unless the non-standard... etc.) That's a simple well-known fact. I'm powerless to change it. > A very basic fact at that: for every term there can be more than one > definition. Depends what you mean by "more then one". Give me another definitions of ds^2 that's not relying on infinitesimals (they are illegal, remember). Other than symmetric rank-2 tensor (or a quadratic form) there is no "more than one definition". > Until you do it is a waste of time to continue discussing this > with you. Just tell me what other definition there is. Just don't use infinitesimals and I'll be happy. -- Jan
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| From | Tom Roberts <tjroberts137@sbcglobal.net> |
|---|---|
| Date | 2016-03-03 14:49 -0600 |
| Message-ID | <tfKdndZl7qz9PkXLnZ2dnUU7_8zNnZ2d@giganews.com> |
| In reply to | #377393 |
On 2/28/16 2/28/16 5:13 AM, Thomas 'PointedEars' Lahn wrote:
> JanPB wrote:
>> It is. Maybe we simply differ in our tolerance to using loose terminology.
Yes, you two do differ on that. As do I.
>> The fact is that ds^2 is a tensor, not a number.
Yes. For the simple reason that JanPB wrote the equations and described his
notation so that ds^2 is a tensor.
But that is not the only interpretation of that notation, nor even the most
common meaning.
In general, one writes
ds^2 = g_ij dx^i dx^j
This can be interpreted TWO COMPLETELY DIFFERENT WAYS:
A) the old/original meaning is that this is a line element, representing
an infinitesimal distance along the line defined by {dx^i} (which are
elementary differentials). So each of the {dx^i} is a (infinitesimal)
real number, as are ds and ds^2 (= (ds)^2); the {g_ij} are a set of
real numbers which are the components of the metric tensor (implicitly
a function of position on the manifold). One must of course integrate
along the line to obtain the square of the distance between any given
pair of points on the line.
B) the modern interpretation as JanPB described: ds^2 is a rank-2 tensor,
d is the exterior derivative, the {x^i} are the usual coordinate
functions on the manifold, the tensor product on the RHS is implicit,
the {g_ij} are a set of real numbers which are the components of the
metric tensor (implicitly a function of position on the manifold), and
"ds^2" is purely a funny name for a tensor to make it LOOK like the
old meaning, with no relationship to either squaring or derivative.
This is a rank-2 tensor, and does not represent distance at all (but
it can be used to determine distance along a specified line).
[In neither case does g_ij represent the metric tensor; it is
the COMPONENTS of the metric tensor.]
In my experience, (A) is far more common than (B). I dislike (B) because of its
rather unusual fudging the notation to make a PUN.
>> And I don't approve of statements like "X represents the geometry" - what
>> does it mean? If X defines the geometry, that's how it should be phrased.
Yes. And X only defines the geometry when X is the metric tensor, because that
_IS_ what defines the geometry [#]. We normally use the symbol g, or g(.,.), or
g_ij for its components, not X.
[#] At least in physics. In mathematics there is more flexibility.
> But “ds” alone does _not_ define the geometry;
See above. ds (A) indeed does not define the geometry (because it is neither the
metric tensor nor its components). But of course the components of the metric
can be easily extracted from the RHS by inspection. Note that ds^2 (B) _IS_ the
metric tensor and DOES define the geometry.
> (c² 0 0 0)
> g_µν = (0 −1 0 0)
> (0 0 −1 0)
> (0 0 0 −1)
> (which – surprise! – *is* also a matrix indeed).
Hmmmm. Yes, the {g_µν} you give are a matrix. They are NOT the metric tensor,
but rather its components. Those components are a set of real numbers which can
of course be converted into the metric tensor g:
g = g_ij dx^i dx^j
where the {dx^i} are the coordinate one-forms (i.e. the exterior derivative d
applied to the coordinate functions {x^i}).
This looks a lot like JanPB's equation, except it avoids
fudging the notation and does not involve a PUN.
In short, you two are arguing about TERMINOLOGY AND NOTATION, not anything of
substance (on which all three of us clearly agree, once the terminological
differences are resolved).
Tom Roberts
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| From | Helénē Swanhild <helenes@gardenclosure.org> |
|---|---|
| Date | 2016-03-03 21:40 +0000 |
| Message-ID | <nbab0c$cee$1@gioia.aioe.org> |
| In reply to | #377818 |
Tom Roberts wrote:
> On 2/28/16 2/28/16 5:13 AM, Thomas 'PointedEars' Lahn wrote:
>> JanPB wrote:
>>> It is. Maybe we simply differ in our tolerance to using loose
>>> terminology.
>
> Yes, you two do differ on that. As do I.
>
>>> The fact is that ds^2 is a tensor, not a number.
>
> Yes. For the simple reason that JanPB wrote the equations and described
> his notation so that ds^2 is a tensor.
>
> But that is not the only interpretation of that notation, nor even the
> most common meaning.
>
> In general, one writes
>
> ds^2 = g_ij dx^i dx^j
>
> This can be interpreted TWO COMPLETELY DIFFERENT WAYS:
> A) the old/original meaning is that this is a line element,
> representing
> an infinitesimal distance along the line defined by {dx^i} (which
> are elementary differentials). So each of the {dx^i} is a
> (infinitesimal) real number, as are ds and ds^2 (= (ds)^2); the
> {g_ij} are a set of real numbers which are the components of the
> metric tensor (implicitly a function of position on the manifold).
> One must of course integrate along the line to obtain the square of
> the distance between any given pair of points on the line.
> B) the modern interpretation as JanPB described: ds^2 is a rank-2
> tensor,
> d is the exterior derivative, the {x^i} are the usual coordinate
> functions on the manifold, the tensor product on the RHS is
> implicit, the {g_ij} are a set of real numbers which are the
> components of the metric tensor (implicitly a function of position
> on the manifold), and "ds^2" is purely a funny name for a tensor to
> make it LOOK like the old meaning, with no relationship to either
> squaring or derivative. This is a rank-2 tensor, and does not
> represent distance at all (but it can be used to determine distance
> along a specified line).
>
> [In neither case does g_ij represent the metric tensor; it is
> the COMPONENTS of the metric tensor.]
You said it was different. Where, they both have the same RHS. They are
not so different, are they, LOL. Same RHS, remember.
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